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Managerial Incentives: On the Near 
Optimal ity of Linearity 

Peter Diamond 

Massachusetts Institute of Technology 

95-6 Jan. 1995 



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Managerial Incentives: On the Near 
Optimality of Linearity 

Peter Diamond 

Massachusetts Institute of Technology 

95-6 Jan. 1995 



OF TECHK'OLOGY 

MAR 2 9 1995 

UBRAHES 



January 31, 1995 



Managerial Incentives: On the Near Optimality of Linearity 

Peter Diamond 

Massachusetts Institute of Technology 



Abstract 

Managers make efforts and choices. Efficient incentives to 
induce effort focus on the signal extraction problem of inferring 
the effort level. Efficient incentives for choices line up the 
relative payoffs of principal and agent. With choices much more 
important than the variation in the cost of inducing effort, the 
optimal payment schedule tends toward proportionality. The 
argument holds if the control space of the agent has full 
dimensionality, but not otherwise. 

If the agent's choices include a complete set of fair gambles 
and insurance, the proportional payoff schedule is no more 
expensive than any other schedule that induces effort. 



mi. 14 

January 31, 1995 

Managerial Incentives: On the Near Optimality of Linearity 
Peter Diamond^ 

Managers are called on to make efforts and to make choices. 
In designing incentives to induce effort, efficiency focuses on the 
signal extraction problem associated with inferring the level of 
effort made (Hart and Holmstrom, 1987) . In contrast, to 
encourage appropriate choices, efficient incentive design tends to 
line up the relative payoffs of principal and agent. The presence 
of choices therefore alters the design of incentives in the standard 
formulation by introducing an additional factor. ^ In many cases 
the choices are much more important than the variation in the 
cost of inducing effort. In this case the optimal payment schedule 
tends toward proportionality.-^ This point is made in Section I in 
a simple model with three states of nature where it is proven that 
as the cost of effort shrinks (relative to gross payoffs) , the 
optimal schedule of payoffs to the agent converges to a linear 
function of gross payoffs. In this three state model, there is a 
single variable describing choice and a single relative payment 
control variable for the principal. 

In Section II, a four state model is used to argue that the same 
logic holds if the control space of the agent has full 
dimensionality, but not otherwise.^ If the principal has more 
degrees of freedom in setting incentives than the agent has 
degrees of freedom in responding, the extra degrees of freedom 
are associated with additional first order conditions that block the 
argument that leads to linearity. The mathematical structure of 
the argument is that the optimal schedule converges to one of the 
optima that induce correct choice when there is no cost of effort. 
When the linear schedule is the unique optimum at zero effort, 
then the optimal schedule converges to a linear schedule. 



1 I am grateful to Daron Acemoglu, Abhijit Banerjee, Oliver Hart, 
Bengt Holmstrom, Jim Mirrlees, Jim Poterba, and participants in 
the Harvard-MIT Theory Workshop for helpful comments and the 
National Science Foundation for research support under grant 
SBR-9307876. 

2 For both choices and efforts, incentives based on outcomes can 
be supplemented by incentives based on observable aspects of 
agent behavior, such as gross inefficiencies. 

3 For other approaches to deriving a linear optimal payment 
schedule, see Holmstrom and Milgrom, 1987, Laffont and Tirole, 
1986, and McAfee and McMillan, 1987. 

4 The dimensionality of the control space is an analog in this 
setup to the "multitask" perspective in Hart and Holmstrom, 1987. 



-1- 



Among the implications of this result is the convergence of 
the optimal schedule to a schedule that ignores the relative 
performance of the firm when information is available on the 
performance of other firms. The needed assumption of full 
dimensionality implies that the agent can trade expected return 
for greater correlation with the returns in the industry. When the 
cost of effort is small, it is better to ignore relative performance 
in order to induce a higher expected return. In the basic model 
presented, effort choice is a zero-one variable. In Appendix B, the 
model is reexamined for the case that effort is a continuous 
variable. 

In Sections I and II, the agent is assumed to be choosing from 
a set of probabilities of different returns that is well-behaved 
relative to the choice problem, but is not otherwise specified. In 
section III, it is assumed that the sets of choices available to the 
agent (with and without effort) include complete sets of fair 
gambles and insurance. In this case a different result is possible. 
If the agent is free to rearrange probabilities across payoffs in 
any way that preserves the expected gross payoff to the principal, 
the proportional payoff schedule is no more expensive than any 
other schedule that induces effort. However, there are many 
schedules that have this property. In section IV, it is assumed that 
the sets of choices available to the agent (with and without effort) 
come from a complete set of fair gambles, but without any 
insurance. Provided one assumes the monotone likelihood ratio 
property, the same conclusion holds as in Section III. 

I Model 

We make the following assumptions. The principal is risk 
neutral. The agent is risk neutral, but can not receive a negative 
payment. There is no individual rationality constraint (the 
expected amount of payment is sufficient to induce supply) . 
There are three states of nature, with payoffs to the principal of 
X3 > 0, X2 > 0, x^ = 0.^^ Assume that the principal chooses three 
payments to the agent satisfying S3 >= 0, S2 >= 0, s^ = 0.^ The 
payment schedule is linear in gross payoffs if S2/S3 = X2/X3. 



5 Adding a constant to these three payoffs makes no change in the 
analysis. 

6 In a one period model, one can not consider the issue of the 
appropriate way to measure the return to the principal. Thus, 
whether payments to managers should depend on some measure of 
accounting profits or current or future stock market values, or 
some combination of these is not addressed. Similarly, the choice 
of a single period schedule of payments to the agent does not 
allow consideration of the use of options as opposed to other 
methods of compensation. 

7 The principal has no reason to use a payment that is the same in 
all states of nature, since it encourages neither more effort nor 
correct choices. Thus this restriction amounts to requiring no 
payment lower than the payment in the state with the lowest gross 
payoff. 

-2- 



Effort 

First we review the standard model, where there is only an 
effort choice. In the standard formulation of this problem, 
expending effort changes the probability vector of the gross 
returns from [l-f2-f3, ±2, £3] to [l-g2-g3, g2/ 93]* With the usual 
formulation, the principal's problem can be stated as 

(1) Maximize (X2-S2)g2 + (X3-S3)g3 

s. t. S2(g2-f2) + S3(g3-f3) >= c, 

Sj^ >= for all i, 

where c is the cost of effort. Given the linearity of this problem 
the optimal scheme can pay compensation in just one state of 
nature. The optimum is found by finding the state or states (other 
than state 1) for which gi/fi is a maximum and paying enough 
compensation in that state to satisfy the incentive compatibility 
constraint. If <32/^2 ~ 93/^3' compensation can be spread across 
both states; otherwise a linear payment schedule is not optimal. 

Effort and Choice 

We modify this model by assuming that expending effort 
generates a set of possible distributions of gross revenues. The 
agent is then free to (costlessly) choose any element in the set. 
This structure is meant to capture the idea that effort generates 
an array of possible strategies for the firm, from which the agent 
selects one - with all strategies requiring roughly equal managerial 
effort to execute. 

To analyze this problem, we work backwards, beginning with 
the agent's choice of the probabilities of payoffs. Denote the 
probability of state two by g. We assume that g is a choice 
variable (within some range) , with h(g) (h'<0, h"<0) being the 
(maximal) induced probability of state three. Over the feasible 
range of values of g, we assume that h' varies sufficiently to 
result in an interior solution. The probability vector of the three 
states is [l-g-h(g), g, h(g)]. Since there is no payoff in state 1, the 
payoff to the agent is S2g+S3h(g) = S3 ( (s2/s3)g+h(g) ) .^ Thus, the 
agent's choice depends only on the ratio of payoffs in states two 
and three which we denote by s=S2/S3. Let us write the chosen 
level of g as g*(s). Then g*(s) maximizes sg+h(g) and is defined by 
the first order condition 

(2) s + h' (g*) = 0. 

Differentiating (2) , we see that g* is monotonically increasing in s. 



8 With the assumption that h is concave, the maximization 
problem of the agent is concave and there is a unique solution. 



-3- 



Conditional on effort being expended, we can write expected 
gross revenue as a function of the relative payoffs, R(s) , as 

(3) R(s) = X2g*(s) +X3h(g*(s)). 

Differentiating (3) and using (2) , we note that R' is zero if and 
only if s gives a linear payoff, s=X2/X3. 

(4) R'(s) = X2g*'(s) + X3h'(g*(s))g*'(s) 

= (X2 - X3S)g*'(s). 

Thus the expected gross revenue, x^g + X3h(g) , is maximized at s 
equal to X2/X3. A deviation from linearity results in a loss in 
expected gross revenue. In this sense, a deviation from linearity is 
similar to a distorting tax. We note that with proportional 
increases in X2 and X3, we have a proportional increase in R' . 

If no effort is made by the agent, the probability vector is [1- 
f2-f3/ f2/ £3]' We assume that it is worthwhile to induce effort. 
Effort can be induced by using payoffs to the agent that give the 
agent an expected return at least as large as the cost of making 
effort, c. 

(5) S2g*(S2/S3) + S3h(g*(S2/S3)) - S2f2 " S3f3 >= C 

Thus we can induce effort with any payoff ratio, s, that satisfies 

(6) sg*(s) + h(g*(s)) - sf2 - f3 > 0. 

That is, for a given ratio, s, that satisfies this inequality, there is 
a value of S3 sufficiently large so that the effort inducement 
constraint (5) is satisfied. For payoff ratios satisfying the 
condition in (6) , by using (5) , the expected cost of just inducing 
effort can be written as: 

(7) C(s) = S2g*(s) + S3h(g*(s)) 

= c[sg*(s) + h(g*(s))]/[sg*(s) + h(g*(s)) - sf2 - ±2]- 
We note that C(s) and C (s) are proportional to c. 

We can now state the optimal incentive problem as 

(8) Maximize R(s) - C(s) . 

We assume that there is a unique internal solution to this problem. 
Then, the first order condition is 

(9) R'(s*) = C'(s*). 

As noted above, preserving the ratio of X2 and X3, R' is 

proportional to X3, while C is proportional to c. Thus, as the ratio 



-4- 



of the cost of effort to gross revenues, C/X3, decreases to zero, 
C'(s)/R'(s) tends to zero unless s converges to X2/X3. Thus, s* tends 
to X2/X3; the optimal incentive structure converges to a linear 
structure. With the cost of inducing effort going to zero (relative 
to gross payoffs) , both s*2/X2 and s 3/X3 tend to zero, but their 
ratio is well-defined, tending to one.' 

Note that the result depends on the ratio C/X3, which can 
become small either by having c become small or by having X3 
become large. For example, if choices affect the profits of a large 
fiirm with profits in the billions, but effort can be induced by 
expected payoffs in the millions, the cost of effort can be large in 
absolute terms, while the ratio of cost to gross payoff is small. 
Note also that the result is that the optimal schedule converges to 
linear, not that the additional cost of a linear schedule becomes 
small relative to X3 (compared with the optimal schedule) . 

II Generalizations 

Four states 

In the three state model analyzed, there is a single variable 
describing choice for the agent and a single relative payment 
control variable for the principal. This structure prevents the 
principal from having additional degrees of freedom for 
encouraging effort beyond the incentive for choice. It is the 
restriction in the dimensionality of agent choice in some 
formulations that results in a basically different structure of 
incentives. To examine this balance between the choice variables 
of the agent and the control variables of the principal we contrast 
two different formulations of the four state problem. 

Now assume that there are 4 states. Assume that the gross 
payoffs satisfy x^ = 0, Xj^ > 0, i=2, 3, 4. Similarly, assume that s-j^ = 
0, Sj^ >= 0, i=2, 3,4. If we assume that the range of choice of the 
agent has full dimensionality (given the constraint that 
probabilities add to one) , then we assume that the agent can 
choose probabilities in two states. That is, the agent can choose 
both g2 and g3 . For any choice in the feasible range (and it is 
assumed that there is a positive range for each variable) there is 
an implied (maximal) level of probability of state 4, written as 
h(g2» 93)* We further assume that the shape of this function is 
such that the optimum involves a unique interior choice. Then we 
have the same first order conditions as before and the argument 



9 If we did not assume risk neutrality for the agent, we would be 
examining the utilities associated with each state rather than just 
the payoff in each state. These utilities might be state-dependent, 
Uj^(sj^), for example if the agent has career concerns and different 
states impacted differently on future opportunities. As long as 
the cost of bearing risk associated with the agent's risk aversion is 
small relative to gross payoffs, we would expect a similar sort of 
convergence for utilities. 



-5- 



goes through in the same way, leading to the conclusion that the 
optimal choices of S2/S4 and S3/S4 converge to X2/X4 and X3/X4. 

In contrast, if we assume that there is a single control 
variable for the agent, g2 , then the probabilities in the other states 
are all functions of this variable, h3(g2) and h4(g2). That is, in the 
case examined just above, the four probabilities are [ l-g2-g3-h(g2 , 
93)/ 92' 93' h(92' 93)3' ^^ '^^® case of a single control variable, the 
probabilities are [l-g2-h3 (g2) "^4 (g2) / 92/ h3(g2), h4(g2)]. In the case 
where the control variables do not span the space of states of 
nature, the principal has another degree of freedom in setting 
payments for the agent, and the argument above does not go 
through. That is, choice depends on the sum S3h3(g2) + S4h4(g2), 
not on the two terms separately. Thus, there is a continuum of 
values of S3 and S4 that result in the same choice and the selection 
of the particular combination of S3 and S4 is made solely to 
minimize the cost of inducing effort. 

Formal argument ^^ 

The contrast between the results with and without full 
dimensionality is striking. This raises the question of the 
underlying mathematical structure for this result. The intuitive 
logic follows; a formal proof is given in Appendix A. As the cost 
of effort goes to zero the set of optimal relative payoffs at each 
cost converges to the set of optimal relative payoffs when the cost 
of effort is exactly zero. If the latter set contains a unique 
optimum, then the optima converge to the linear schedule, which 
is always an optimum with no cost of effort. However, if the set 
of optima at zero cost contains other solutions besides the 
proportional schedule, then the convergence to the set of optima 
may not involve convergence to the linear schedule. The set of 
optima with zero cost is the set of supports to the set of available 
distributions at the point that maximizes expected gross payoffs. 
With full dimensionality and a lack of kinks, the support for the 
set of distributions at the point that maximizes expected gross 
returns will be unique. Then the argument in the text goes 
through. However, without full dimensionality, there is not 
uniqueness in the set of supporting prices and convergence will 
not generally hold. 

Relative performance 

The model can be extended to a situation where information 
is available on the performance of other firms in the industry. 
The critical question remains that of dimensionality - whether the 
agent has the ability to modify the correlation between the returns 
to the principal and the returns to other firms. When full 
dimensionality remains present, the linearity argument goes 
through, implying that the optimal incentive scheme converges to 
one that ignores relative performance. 



10 This argument was made by Jim Mirrlees. 



-6- 



To review this formally, assume that the gross payoff to the 
principal can be high or low and the gross payoff to the industry 
can be high or low. Thus there are four states, given by the two- 
by-two matrix of low and high returns for the principal and the 
industry. With a zero payment in the event that the principal's 
payoff is low while the industry payoff is high, the principal has 
three controls, or two relative controls. If the agent's action space 
is of full dimensionality, the agent can choose two payoff 
probabilities. In this case the argument above goes through, with 
the same result, that in the limit the payoff schedule should 
ignore the performance of the industry. The problem with trying 
to use such information to hold down the cost of inducing effort 
is that it affects the expected gross payoff as the agent responds 
to the incentive structure by altering the correlation of the 
principal's payoff with that of the industry. 

Perks 

The model above does not have explicit representation of 
decisions by the agent that directly affect the agent's utility at the 
cost of expected gross returns. Excessive numbers of limousines 
and jets are representative examples of such actions. More 
expensive are new corporate headquarters, or relocation of 
corporate headquarters for the pleasure of the agent. Even more 
expensive might be empire building. Since there is no limit on the 
ability to waste and since it is inevitable that top management 
will have only a small share of the variation of returns in a large 
firm, the presence of such possible actions calls for attempts to 
monitor them directly. Thus, just as the principal sets the cash 
reimbursement schedule of the agent and needs to check that the 
agent does not receive more cash compensation than the agent is 
entitled to, so the principal needs to monitor the noncash 
reimbursement of the agent. With sufficiently effective 
monitoring, the argument above should remain, so that 
complicated reimbursement schedules, as called for by the model 
with only effort, remain unnecessary, and possibly ineffective. 

Another decision that involves large returns is that of 
replacement of the agent by another agent. The model above has 
not considered such a possibility, one that complicates the analysis 
by having nonfinancial returns associated with replacement. 

Ill Fair gambles and insurance 

In the usual formulation of the principal-agent problem, (1) , 
the optimal schedule selects the state (or states) for which gi/fi is a 
maximum and pays enough compensation in that state to induce 
effort. If state i is the lowest cost state, then, paralleling the 
argument leading to (7), the cost of inducing effort is cgj^/ (g^^-f ^ ) = 
c(gi/fi) / ( (gi/fi)-l) . Note that the function cz/(z-l) is decreasing in 
z. Thus, the principal rewards the agent in the state where gi/fi is 
largest. Only if g2/f2 ~ 93/^3/ can the optimum include 
compensation in both states. 



-7- 



In this formulation, expending effort changes the probability 
vector of the gross returns from [l-f2-f3/ fp, f3] to [l-g2~g3/ <32i ^3) 
In the analysis above, we added a choice variable that was only 
available if effort was expended. We now consider the case where 
choice is available whether or not the agent expends effort. 
However, we now assume that the choice set of the agent is 
defined by the ability to take all possible fair gambles and fair 
insurance policies (or hedging possibilities) . That is, we assume 
the agent can rearrange the probabilities of the different states in 
any way that preserves the expected gross payoff to the principal. 

Proposition. Assume that there are n states, with 
0=X2^<X2<X3<. . .<Xj^. Assume that whether effort is exerted or not, 
the agent has access to all fair gambles and insurance. Then the 
linear schedule costs no more than any other schedule. 

We proceed by first examining schedules that pay the agent in 
just one state, then considering all alternatives. 

Payoff in one state 

Let us denote the mean gross payoffs without and with effort 
by m^ and mg. With a proportional payoff schedule, gambles that 
do not change the expected return to the principal, do not change 
the expected payoff to the agent. Thus we can ignore fair 
gambles in evaluating the proportional schedule and note, from 
the argument leading to (7) , that the cost of inducing effort with 
the proportional schedule is 

(10) C = cmg/(mg-mf) = c(mg/mf ) / ( (mg/mf ) -1) . 

Next consider a payoff schedule that gives compensation only 
in state i. The agent wants to concentrate as much probability as 
possible on this state. If x-j^ exceeds the mean return, the agent 
maximizes the expected payoff by taking gambles so that all 
probability is concentrated on states i and 1. If Xj^ is less than the 
mean return, the agent maximizes the expected payoff by taking 
gambles so that all probability is concentrated on states i and n. 
Let us denote the probabilities of payoffs after such gambles by f 
and g' in the cases that effort is not and is expended. Then we 
have two cases (ignoring the state with a payoff precisely equal to 
the mean, in which case the probability of such a state can be set 
to 1) . 

(11) If Xj^ > mf, then f'j^ = m^/Xj^. 

If Xi < mf, then f'^ = (x^-mf ) / (x^-x^) . 

The same rule holds for g. 

To induce effort using a schedule that pays A in state i, we 
note that A must satisfy 

(12) A(g'i-f'i) = c. 



-8- 



Thus, the expected cost satisfies 

(13) C = Ag'i = cg'i/(g'i-f' i) = c(g'i/f' i) / ( (g'i/f i)-l) . 

We noted that the cost, C, is decreasing in q' /t' . 

We have three cases as Xj^ exceeds nig, lies between m^ and lUg, 
and is less than nif. In these three cases, we have: 

(14) If Xi > mg > mf, g'i/f'i = nig/nif. 

If mf < Xi < lUg, g'i/f'i = (Xn-ing)Xi/(Xn-Xi)inf < mg/itif. 

If Xi < nif < lUg, g'i/f'i = (Xn-ing)/(Xn-inf) < mg/mf. 

Comparing (13) and (10) and using (14) , we conclude that costs are 
at least as high with payoff to the agent in a single state as with 
proportional payoffs. In the first case, where the payoff is in a 
state with high return, we have the same cost for the schedule 
paying in the single state as for the schedule with proportional 
payoffs. In the other two cases, costs are higher with the payoff 
in a single state than with proportional payoffs. Thus, with fair 
gambles, the agent is relatively better able to take advantage of 
gambles when there is the lower expected return without effort. 
For example, in the case Xj^ < m^ < mg, the ratio of probabilities of 
collecting, g'i/f'i, converges to one as Xj^ rises without limit. The 
convergence of the probabilities makes it expensive to induce 
effort with this schedule. 

General case 

Schedules that pay the agent in just one state are at least as 
expensive as the proportional schedule. We extend the argument 
to more complicated schedules using the following argument. First 
we argue that the linearity of this problem implies that an 
optimum can be found where the agent puts probability weight on 
no more than two states. In turn, this implies that costs are not 
increased and effort not discouraged by setting payoffs equal to 
zero in states in which the agent puts no probability if effort is 
taken. An argument directly comparing costs, parallel to that 
above, completes the proof. 

We begin with the agent's problem after exerting effort: 

(15) Maximize q' 2^2"^^' 2^2'^' • '^^' n^n 

subject to g'2X2+g'3X3+. . .+g'n^n = mg, 

g'j^ >= for all i. 

This is a linear programming problem. We can conclude that an 
optimum can be found by the agent in which probability is put on 



-9- 



no more than two states. In order to preserve the mean payoff to 

the principal, these two states, i and j, say, must lie on either side 

of m„: 

(16) Xi < mg < Xj. 

Denote the probability placed on state i by g' , with the probability 
on state j being 1-g'. Then, from the constraint on expected gross 
returns, we have 

(17) g' = (Xj-mg)/(Xj-Xi). 

The mean return without effort, m^, might lie in the interval 
between x^ and x j , or might be less than Xj^. In the former case, 
denoting the no-effort probabilities by f and l-f, we have: 

(18) f = (Xj-mf)/(Xj-Xi). 

In this case, the cost of just inducing effort is 

(19) Sj - (Sj-Si)g', 
where 

(20) (Sj-Si) (f'-g') = c. 

Thus the cost of just inducing effort using payoffs only on states 
i and j is 

(21) Sj - (Sj-Si)g' = Sj - cgV(f'-g'). 

This is minimized at Sj = 0, implying we are back in the case with 
payment in a single state, which does not cost less than the 
proportional schedule. 

In the remaining case, m^ < Xj^. If Sj^/Xi < sa/xa, after exerting 
effort, the agent will do at least as well using states 1 and j rather 
than states i and j. But this involves a payoff in just one state, 
and does not cost less than the proportional schedule. Thus we are 
left with the case that there are positive payoffs in two states and 
the state with the lower gross payoff has at least as high a relative 
payoff as the state with the higher gross payoff. Note that with 
effort, the agent puts probability on states i and j, while without 
effort, the agent puts probability on states 1 and i. In this case, 
we have 



(22) f = mf/Xi 



1' 



while g' satisfies (18) . The cost of effort is equal to the 
difference between expected returns with and without effort, 
while the expected cost to the principal of inducing effort is the 
expected return with effort, which equals the cost of effort plus 
the expected return without effort. 



-10- 



(23) C = c + Sj^mf/Xj^. 

Thus lowering s^ while raising Sj to preserve the incentive for 
effort lowers expected cost to the principal. This argument holds 
as long as the relative payoff is higher on state i. Thus we have 
not raised the cost to the principal by going to a schedule with the 
same relative payoffs. But this case is covered by the argument 
just above. 

Thus we can conclude that when the set of alternative choices 
available to the agent is the set of all fair gambles and hedges, the 
proportional payoff schedule does at least as well as any other 
schedule. Note that in this case, there are many equally good 
schedules. The linear schedule is one of these, but so, too, are the 
schedules that pay in just one state of nature, provided that the 
state with the payoff has a gross return at least as large as the 
mean return when making effort. 

IV Fair Gambles and the Monotone Likelihood Property 

In the argument above, we have assumed no restriction on the 
ability of the agent to move probability across the states of nature 
other than preserving the mean. This assumption might be viewed 
as a rich set of alternative investments that have a frontier with 
this property. Alternatively, the ability to move probability can 
be viewed as coming from the availability of both fair gambles 
and fair insurance. There is an asymmetry between gambles and 
insurance in that the former can be done after the agent knows 
the realized before-gamble return, while the latter requires ex ante 
arrangements. Thus it is natural to consider the case where the 
agent can take any fair gamble, but has no insurance available at 
all. That is, we assume that decreasing the probability in some 
state requires increasing the probabilities in states with both 
higher and lower returns. This implies that probability can be 
moved out of any state except the ones with the highest and 
lowest returns. 

With just this assumption, it is not the case that the linear 
schedule can do as well as any other schedule. Consider a three 
state model where without the availability of gambles, the unique 
optimum is to pay the agent only in state 2. Then, with this 
payoff schedule, the availability of gambles does not alter what 
the agent will do, with or without effort. Thus, this remains the 
unique optimal schedule. To analyze this situation further, we 
also assume the monotone likelihood ratio property. ^-^ 

Proposition. Assume that there are n states with 
0=X]^<X2<X3<. . .<x,^. Assume that all states have positive probability 
without and with effort: fi>0, gi>0. Assume that 



11 For analysis of this problem without gambles, see Innes, 1990. 



-11- 



gi/fl<g2/f2<g3/^3"^* • '"^^n/fn* Assume that whether effort is exerted 
or not, the agent has access to all fair gambles, but no insurance. 
Then the linear schedule costs no more than any other schedule. 

The proof proceeds in three steps. First we observe that the 
linear schedule costs exactly the same as the schedule that pays 
only in state n. Second, we show that the schedule paying only in 
state n is strictly better than the schedule paying only in one state 
other than state n. Third, we argue that any schedule with a 
relative payoff in some state higher than that in state n can be 
improved upon. 

With a payoff only in state n, the absence of insurance is of 
no consequence and the argument is the same as that used above. 
Similarly, we note that any schedule that has no state with a 
higher relative payoff than in state n costs the same as the 
schedule with payoff only in state n. This follows from the fact 
that with such a schedule, the agent would shift all probability to 
states 1 and n. 

Now consider the schedule with a payoff only in state i>l. 
The agent will take gambles so there is no probability on any state 
between 1 and i. Thus, the probabilities of receiving payoffs, f'j^ 
and g'j^ satisfy 

(24) f i = (fiXi+f2X2+f3X3+...fiXi)/Xi, 

g'i = (giXi+g2X2+g3X3+...+giXi)/xi. 

Thus we see that g'^/f'^ is increasing in i. Since the cost is 
decreasing in g'j^/f'j^, as we saw above, this completes the second 
step in the proof. 

Now consider an arbitrary payoff schedule which has a 
higher relative payoff in some state than in state n. Let state i be 
the state with the highest index (highest gross payoff) that has a 
higher relative payoff than does state n, Sj^/Xj^ > Sj^/x^. Since 
relative payoffs on states between i and n are lower than those at 
i and n, gambles are taken until there is no probability on states 
between i and n. For any state with index below i, none of the 
probability is moved to state n. These statements hold both with 
and without effort. Thus, there is more probability placed on 
state n with effort than without effort. Consider slightly raising 
Sj^ and lowering all other payoffs, with the decreases in other 
payoffs proportional to gross payoffs, Xj^, and done so that the 
payoff with effort is unchanged. The payoff to gambles not 
transferring probability to state n is unchanged. Thus there are 
no changes in gambles, and the inducement to effort now exceeds 
the cost of effort, allowing a reduction in all payoffs, and so in 
the cost of the schedule. 

From the argument, we see that all schedules for which the 
relative payoff in every state is no larger than the relative payoff 
in the highest state have the same cost. This includes the linear 
schedule. This observation completes the proof. 



-12- 



V Concluding remarks 

In many settings, the cost of inducing managers to work hard 
is far less important than encouraging them to make the right 
choices from the set made available by their hard work. For 
example, the managers of large firms have effort costs which are 
very small compared to the range of possible profits of the firms, 
which can be in the billions. In modelling this situation, it was 
assumed that agents were risk neutral, but could not be paid a 
negative amount. This structure was designed to capture several 
aspects of large firms. One is that the wealth of management is 
indeed small relative to the value of the firm. Thus there is no 
way that the payoff to management can vary as a significant 
fraction of gross payoffs to the firm without having very large 
expected payoffs to the management. Thus the focus is on the 
structure of payoffs to the agent relative to gross payoffs, not the 
level of the ratio. Convergence of the payoffs to the agent to 
zero, relative to gross payoffs, is an implication of large firms 
without a large enough set of suitably wealthy individuals. 

One assumption of the basic model is that agents make 
optimal choices. If the variation in returns were sufficiently 
small, agents might not bother making optimal choices, since 
choosing might not be costless, as was assximed. The convergence 
of the optimal schedule to one that is linear with positive slope, 
rather than to a flat salary, makes this concern not seem 
important . 

Assuming the agents are well-enough paid in the worst state to 
be risk neutral reflects two presumed aspects of a more basic 
model. One is that it is probably efficient for a large firm to 
absorb the risk of agents as a way of holding down the cost of 
inducing a supply of suitable agents. Second is that the risk 
aversion of agents will lead to poor choices for a risk neutral 
principal. Thus significant minimal payoff takes care of the risk 
aversion problem (except that of losing the job) at relatively small 
cost to the firm. Given the assumptions made, a proportional 
payoff schedule (in utilities) will be nearly optimal. If, as also 
assumed, managers are paid sufficiently well to be risk neutral, 
then a proportional payoff schedule will be nearly optimal. 

The second argument made in the paper is that the ability of 
managers to alter the probability structure of expected payoffs 
gives management an opportunity to take advantage of some 
nonlinear ities in the payment schedule, implying that a nonlinear 
structure would result in higher expected costs to the principal. 
Thus, when the ability to manipulate probabilities is large enough, 
the proportional schedule is optimal even if the cost of inducing 
effort is significant. 



-13- 



Appendix A: Proof of convergence-^^ 

The argument in the text for the convergence of the optimal 
schedule to a linear schedule was made in calculus terms in a 
model with three states of nature. We now sketch a noncalculus 
proof of the same argument, without the restriction to three states 
of nature. We will see that the argument goes through when there 
is a unique revenue maximizing schedule, but not necessarily 
otherwise. We assume that it is worthwhile to induce effort. 

Assume n states of nature, and write as G the n-1 dimensional 
choice set (omitting state 1) of the agent, assuming effort. We 
assume that G is a convex set. Let us define C(s) as the cost to 
the principal of the optimal choice of g by the agent in the 
presence of a reward schedule s: 

Al C(s) = Maximiam g.s for g in G. 

Note that C(s) is homogeneous of degree 1 in s. We define G(s) as 
the set of probabilities that achieve this maximum: 

A2 G(s) = { g I g.s = C(s) }. 

We note that the set G(s) is homogeneous of degree zero in s. 

Similarly, we define the gross revenue generated by a reward 
schedule, s, assuming that indifference on the part of the agent 
results in a choice that maximizes gross revenue for the principal. 

A3 R(s) = Maximum g.x for g in G(s) . 

Note that R(s) is maximized over s when s is proportional to x 
since G(s) is a subset of G and maximal revenue over G comes 
from a choice in G(x) . 

Define as S the set of rewards that result in selection of the 
revenue maximizing probabilities over the entire set G. 

A4 S = { s I R(s) = R(x) } 

We note that the set S is homogeneous of degree zero in s. If G 
has a unique supporting hyperplane at the revenue maximizing 
probabilities, then S contains only the point x and its scalar 
multiples. If G is not of full dimension, then there will be 
additional points in S. 

We can now write the principal's problem as choosing g and s 
to: 

A5 Maximize g.x - C(s) 



12 This approach and proof were provided by Jim Mirrlees. 



-14- 



subject to C(s) - f.s = c, g in G(s) , s>=0. 

Equivalently, ,we can consider the problem of choosing relative 
rewards, s, constrained to add to one, and a scalar multiple, z. 
Thus we restate the principal's problem as choosing g, s and z to: 

A6 Maximize g.x - zC(s) 

subject to z[C(s) - f.s] = c, g in G(s) , s>=0, l.s = 1. 

Using the moral hazard constraint to eliminate z, we can restate 
the principal's problem when c>0 as choosing g and s to: 

A7 Maximize g.x - cC(s)/[C(s) - f.s] 

subject to C(s) > f.s, g in G(s), s>=0, l.s = 1. 

In order to make the limit argument, we now consider a 
sequence of solutions to this problem as c varies. Consider a 
sequence of values of c, Cj^, which converges to zero. For each Cj^, 
select a pair of values of g and s, gj^ and Sj^, which maximize the 
principal's net revenue, that is solve A7. Next consider a 
subsequence of values of c (retaining the same notation) for which 
the subsequence Sj^ converges to some value, named t. Next, 
consider a subsequence of this subsequence (retaining the same 
notation) for which the subsequence gj^ converges to some value, 
named h. We will have completed the proof if we show that t is 
in S. That is, we will have shown that as c decreases to zero, 
every convergent sequence of optimal rewards converges to a point 
in S. If there is a unique supporting hyperplane to G at the 
revenue maximizing point, then the limit point t is proportional to 

X. 

Lemma 1. Lim sup R(Sj^) <= R(t) . 

From the optimality of Sj^ and gj^, we have g^.Sj^ >= g.Sj^ for all g 
in G. Passing to the limit we have h.t >= g.t for all g in G. Thus 
we have h in G(t) , implying that h.x <= R(t) . With Sj^ converging 
to t, we have g^.x converging to h.x, completing the proof. 

Lemma 2. Lim sup R(Sj^) >= R(x) . 

Let X' be proportional to x and satisfying l.x'=l. From the 
optimality of Sj^ and g^, we have R(S4) - cC(Sj^) / [C(Sj^) - f.Sj^] >= R(x') 
cC(x' ) / [C(x') - f.x']. Taking the limit as c goes to zero completes 
the proof. 

Thus we have shown that R(t) = R(x) , implying that s^ 
converges to one of the points in S. If G has a unique supporting 
hyperplane, then Sj^ converges to proportionality with x, that is, t is 
a linear payoff schedule. If G does not have a unique support, 
because of a kink or less than full dimensionality, then the limit 
of Sj^ need not be proportional to x. 



-15- 



Appendix B: Continuous effort 

The discussion above was made simpler by the discrete nature 
of the effort choice. In this section, I briefly turn to the case of a 
continuous adjustment of effort. Let us denote effort by e and its 
cost to the agent by ce. First we review the problem if there is no 
choice variable, only an effort variable. In this case, we write the 
probabilities as functions of effort, hj^(e). In this setting, the agent 
maximizes S2h2(e) + S3h3(e) - ce. This gives the first order 
condition for the agent's choice 

Bl S2h'2 + ^3^'3 ~ ^' 

From the first order condition, we can write the chosen level of 
effort as a function of the payments, e*(s2, S3). We note that the 
derivatives of e* with respect to payments are proportional to the 
values of h' : 

B2 e*i = -h'i/(S2h"2 + S3h"3). 

We can now state the principal's problem as 
B3 Max (X2 - S2)h2(e*(S2, S3)) + (X3 - S3)h3(e*(S2, S3)). 
The first order conditions for the optimal incentive schedule are 
B4 -h2 + (X2-S2)h'2e*2 "•" (X3"S3)h'3e*2 = 0, 

-h3 + (X2-S2)h'2e*3 + (X3-S3)h'3e*3 = 0. 
Using (Bl) , we can rewrite the first order conditions, (B4) , as: 
B5 -h2 + (X2h'2 + X3h'3 - C)e*2 = 0, 

-h3 + (X2h'2 + ^3h'3 ~ c)e*3 = 0. 
or, using (B2) : 
B6 -h2(S2h"2 + S3h"3) + (X2h'2 + X3h'3 - c)^' 2 ^ °' 

-h3(S2h"2 + S3h"3) + (X2h'2 + X3h'3 " ^)^' 3 = 0* 

Thus, unless h'j^/hj^ is the same for both states, the optimum offers 
a payment in only one state of nature. This is similar to the 
situation with a discrete choice of effort level. 

Now let us assume that the probability of state three is a 
function of both the effort undertaken and the choice of 
probability of state two, h(g,e).^-^ In this setting, the agent 



13 We are ignoring the impact of effort on the available range of 
values of g. 



•16- 



maximizes s^g + 3311 (g,e) - ce. This gives the pair of first order 
conditions for the agent's effort and choice: 

B7 S3hg = c, 

S2 + S3hq =0. 

From the first order conditions, we can write the choice variables 
as functions of the payments, g (S2, S3) and e*(S2, S3). 

We can now state the principals problem as 

B8 Max (X2 - S2)g*(S2, S3) + (X3 - S3)h(g*(S2, S3), e*(S2, S3)). 

The first order conditions for the optimal incentive schedule are 

B9 -g* + (X2-S2)g*2 + (X3-S3) (hgg*2+hee*2) = 0, 

-h(g*,e*) + (X2-S2)g*3 + (X3-S3) (hgg*3+hee*3) = 0. 

Following the same tack as previously, let us group the first order 
conditions between terms relating to choice and those relating to 
effort: 

BIO X2g*2 + X3hgg*2 = g* + S2g*2 + S3(hgg*2+hee*2) - X3hee*2, 

^29*3 + X3hgg*3 = h(g*,e*) + S2g*3 + S3 (hgg*3+hee*3) - X3hee*3. 

Now using the first order conditions for the agent's problem, we 
can write this as: 

Bll (X2 - X3(S2/S3))g*2 = g* + S3hee*2 - X3hee*2, 

(X2 - X3(s2/S3))g*3 = h(g*,e*) + S3hee*3 - X3hee*3. 

or 

B12 (X2 - X3(S2/S3))g*2 = g* + (S3 - X3)hee*2, 

(X2 - X3(s2/S3))g*3 = h(g*,e*) + (S3 - X3)hee*3. 

If the right hand sides of these two equations go to zero, then 
S2/S3 converges to X5/X3. But the right hand sides of equations 
(B12) are precisely the first order conditions for the agent's 
efforts in the model without choice, equation (B4) . So, we have a 
similar conclusion to that above - when the variation in the cost 
of inducing optimal effort becomes small relative to gross payoffs, 
the incentive schedule converges to linear. 

The exploration of sufficient conditions for the effort choice 
not to be important at the margin is more complicated when effort 
is a continuous variable than when it is a zero-one variable, and I 
will not examine it in detail. In the latter case, it was sufficient 



-17- 



to have the cost of effort become small relative to gross payoffs, 
C/X3 go to zero. In this case, one needs to have assumptions that 
limit the increase in effort, since ce/X3 need not go to zero when 
C/X3 does. Thus, we would want to make the plausible assumption 
that hg goes to zero at a finite level of e. We might also want to 
rule out the possibility that changes in effort have important 
effects on the slope of the tradeoff between probabilities in both 
states. That is, we might also want to assume that hgg goes to zero 
as well. 



-18- 



References 

Hart, Oliver, and Bengt Holmstrom, 1987, The theory of contracts, 
in T. Bewley (ed.) , Advances in Economic Theory . Cambridge, 
Cambridge University Press. 

Holmstrom, Bengt, and Paul Milgrom, 1987, Aggregation and 
linearity in the provision of intertemporal incentives, 
Econometrica . 55, 303-28. 

Innes, Robert D., 1990, Limited liability and incentive contracting 
with ex-ante action choices, Journal of Economic Theory . 52, 1, 
45-67. 

Laffont, Jean-Jacques, and Jean Tirole, 1986, Using cost 
observation to regulate firms. Journal of Political Economy . 94, 
614-641. 

McAfee, R. Preston, and John McMillan, 1987, Competition for 
agency contracts, RAND Journal of Economics . 18, 296-307. 



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